Self-orthogonal flags of codes and translation of flags of algebraic geometry codes

Abstract

A flag $C_0⊊C_1\cdots ⊊C_s⊊{\mathbb{F}}_q^n$ of linear codes is said to be self-orthogonal if the duals of the codes in the flag satisfy $C_i^{\perp }=C_{s-i}$, and it is said to satisfy the isometry-dual property with respect to an isometry vector $\mathbf{x}$ if $C_i^{\perp }=\mathbf{x}{C}_{s-i}$ for $i=1,\dots ,s$. We characterize complete (i.e. $s=n$) flags with the isometry-dual property by means of the existence of a word with non-zero coordinates in a certain linear subspace of ${\mathbb{F}}_q^n$. For flags of algebraic geometry (AG) codes we prove a so-called translation property of isometry-dual flags and give a construction of complete self-orthogonal flags, providing examples of self-orthogonal flags over some maximal function fields. At the end we characterize the divisors giving the isometry-dual property and the related isometry vectors showing that for each function field there is only a finite number of isometry vectors and that they are related by cyclic repetitions.